Finite Group Databasebeta

← C3×#10595049 #13174639

Subgroups — C3×#10595049 #13174639

Order 1728.

1202 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64135C211yesmaximal, characteristic
2C32×#926C211yesmaximal, characteristic
3C2×C32×C4×Dic3C211yesmaximal, characteristic
4C3×#63964C214yesmaximal
5C3×#63918C212yesmaximal, complemented
6C3×C4×#627C214yesmaximal, complemented
7C3×#63869C212yesmaximal, complemented
8C3×#5517—31nomaximal
9C3×#5470—31nomaximal
10#10595049C311yesmaximal, characteristic, direct product factor
11C22×C33×C4C2211yescharacteristic
12C22×C32×Dic3C2212yes—
13C2×C3×#627C2214yes—
14C3×#3757C2212yes—
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×C4×C3⋊Dic3C2212yes—
17C32×#217C2212yes—
18C32×C4⋊Dic3C2212yes—
19C32×C4×Dic3C2211yescharacteristic
20C32×C4×Dic3C2212yes—
21C32×C4×Dic3C2211yescharacteristic
22C32×C4×Dic3C2212yes—
23C3×#3724C2214yes—
24C3×Dic3⋊Dic3C2214yes—
25C3×Dic3⋊Dic3C2214yes—
26C32×#117—31no—
27C2×C32×C42—31no—
28#64135C2×C311yescharacteristic
29C3×#926—21no—
30C3×#926—31no—
31C3×#926—31no—
32C3×#926—61no—
33C3×#926D311yescharacteristic
34C3×#926C2×C311yescharacteristic
35C3×#924—34no—
36C2×C3×C4×Dic3—21no—
37C2×C3×C4×Dic3D311yescharacteristic
38C2×C3×C4×Dic3C2×C311yescharacteristic
39C3×#890—32no—
40C3×#888—32no—
41C3×#878—34no—
42C3×#870—32no—
43C3×#869—32no—
44C3×C4×Dic6—34no—
45C3×C4×Dic6—34no—
46#63964C2×C314yes—
47#63918C2×C312yescomplemented
48C4×#627C2×C314yescomplemented
49#63869C2×C312yescomplemented
50C23×C33C2311yescharacteristic
51C2×C33×C4Q814yescomplemented
52C2×C33×C4C2311yescharacteristic
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3C2314yes—
55C2×C32×Dic3—24no—
56C2×C32×Dic3C2314yes—
57C2×C32×Dic3C2312yes—
58C2×C32×Dic3C2312yes—
59C3×#627—28no—
60C3×#533—91no—
61#5517—31no—
62#5470—31no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4—32no—
67C22×C32×C4—41no—
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C2×D311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C3×Dic3—22no—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C22×C312yes—
74C2×C3×Dic6—34no—
75C2×C3×Dic6—34no—
76C2×#627C22×C314yes—
77C32×C4⋊C4—32no—
78C32×C22⋊C4—32no—
79C32×C42—32no—
80C32×C42—31no—
81C32×C42—31no—
82C32×C42—32no—
83#3757C22×C312yes—
84C4⋊(C3⋊Dic3)C22×C312yes—
85C4×C3⋊Dic3C22×C312yes—
86C3×#217—22no—
87C3×#217—32no—
88C3×#217—32no—
89C3×#217—62no—
90C3×#217C2×D312yes—
91C3×#217C22×C312yes—
92C3×C4⋊Dic3—22no—
93C3×C4⋊Dic3—32no—
94C3×C4⋊Dic3—32no—
95C3×C4⋊Dic3—34no—
96C3×C4⋊Dic3—34no—
97C3×C4⋊Dic3—62no—
98C3×C4⋊Dic3C2×D312yes—
99C3×C4⋊Dic3C22×C312yes—
100C3×Dic3⋊C4—34no—
101C3×Dic3⋊C4—34no—
102C3×Dic3⋊C4—34no—
103C3×Dic3⋊C4—34no—
104C3×C4×Dic3—22no—
105C3×C4×Dic3—21no—
106C3×C4×Dic3—21no—
107C3×C4×Dic3—22no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—62no—
111C3×C4×Dic3C2×D311yescharacteristic
112C3×C4×Dic3C2×D311yescharacteristic
113C3×C4×Dic3C2×D312yes—
114C3×C4×Dic3C2×D312yes—
115C3×C4×Dic3C22×C311yescharacteristic
116C3×C4×Dic3C22×C312yes—
117C3×C4×Dic3C22×C311yescharacteristic
118C3×C4×Dic3C22×C312yes—
119#3724C22×C314yes—
120Dic3⋊Dic3C22×C314yes—
121Dic3⋊Dic3C22×C314yes—
122C22×C33—22no—
123C22×C33C2×Q812yes—
124C22×C33C2411yescharacteristic
125C33×C4C2×Q812yes—
126C33×C4C2×Q812yes—
127C3×C3⋊Dic3—24no—
128C32×Dic3—22no—
129C32×Dic3—24no—
130C32×Dic3#4114yes—
131C3×C4⋊Q8—92no—
132C3×#125—92no—
133C3×C22⋊Q8—94no—
134C3×C4×Q8—94no—
135C3×#117—31no—
136C3×#117—31no—
137C3×#117—61no—
138C3×#117—91no—
139C2×C3×C42—31no—
140C2×C3×C42—31no—
141C2×C3×C42—61no—
142#926—31no—
143#926—31no—
144#926—61no—
145#926C3×D311yescharacteristic
146#924—34no—
147C2×C4×Dic3C3×D311yescharacteristic
148#890—32no—
149#888—32no—
150#878—34no—
151#870—32no—
152#869—32no—
153C4×Dic6—34no—
154C4×Dic6—34no—
155C23×C32—21no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32—41no—
159C23×C32C22×D311yescharacteristic
160C23×C32C22×D311yescharacteristic
161C23×C32C23×C311yescharacteristic
162C2×C32×C4—24no—
163C2×C32×C4—24no—
164C2×C32×C4—24no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4—21no—
170C2×C32×C4—21no—
171C2×C32×C4—32no—
172C2×C32×C4—32no—
173C2×C32×C4—34no—
174C2×C32×C4—44no—
175C2×C32×C4—41no—
176C2×C32×C4—41no—
177C2×C32×C4—64no—
178C2×C32×C4Dic614yescomplemented
179C2×C32×C4Dic614yescomplemented
180C2×C32×C4C3×Q814yescomplemented
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C23×C311yescharacteristic
186C2×C32×C4C23×C311yescharacteristic
187C2×C3⋊Dic3C23×C314yes—
188C2×C3×Dic3—28no—
189C2×C3×Dic3—24no—
190C2×C3×Dic3—24no—
191C2×C3×Dic3—22no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—24no—
194C2×C3×Dic3—34no—
195C2×C3×Dic3—34no—
196C2×C3×Dic3—64no—
197C2×C3×Dic3C22×D312yes—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D312yes—
200C2×C3×Dic3C23×C312yes—
201C2×C3×Dic3C23×C314yes—
202C2×C3×Dic3C23×C312yes—
203C3×Dic6—68no—
204C3×Dic6—68no—
205#627—28no—
206#533—91noSylow
207C2×C33—22no—
208C2×C33C22×Q811yescharacteristic
209C2×C33C2×#4112yes—
210C2×C3×Q8—94no—
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—32no—
215C22×C3×C4—32no—
216C22×C3×C4—41no—
217C22×C3×C4—62no—
218C22×C3×C4D3211yescharacteristic
219C22×C3×C4C2×C3×D311yescharacteristic
220C22×C3×C4C2×C3×D311yescharacteristic
221C22×Dic3C2×C3×D312yes—
222C2×Dic6—34no—
223C2×Dic6—34no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—32no—
226C3×C4⋊C4—62no—
227C3×C4⋊C4—94no—
228C3×C4⋊C4—94no—
229C3×C4⋊C4—94no—
230C3×C4⋊C4—92no—
231C3×C22⋊C4—32no—
232C3×C22⋊C4—32no—
233C3×C22⋊C4—62no—
234C3×C22⋊C4—92no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42—32no—
238C3×C42—32no—
239C3×C42—32no—
240C3×C42—32no—
241C3×C42—31no—
242C3×C42—31no—
243C3×C42—62no—
244C3×C42—61no—
245C3×C42—61no—
246C3×C42—62no—
247C3×C42—92no—
248#217—32no—
249#217—32no—
250#217—62no—
251#217C2×C3×D312yes—
252C4⋊Dic3—32no—
253C4⋊Dic3—32no—
254C4⋊Dic3—34no—
255C4⋊Dic3—34no—
256C4⋊Dic3—62no—
257C4⋊Dic3C2×C3×D312yes—
258Dic3⋊C4—34no—
259Dic3⋊C4—34no—
260Dic3⋊C4—34no—
261Dic3⋊C4—34no—
262C4×Dic3—32no—
263C4×Dic3—32no—
264C4×Dic3—62no—
265C4×Dic3C2×C3×D312yes—
266C4×Dic3C2×C3×D312yes—
267C4×Dic3C2×C3×D311yescharacteristic
268C4×Dic3C2×C3×D311yescharacteristic
269C22×C32—22no—
270C22×C32—24no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—21no—
274C22×C32—21no—
275C22×C32—21no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—22no—
279C22×C32—42no—
280C22×C32—41no—
281C22×C32—42no—
282C22×C32—42no—
283C22×C32—44no—
284C22×C32C2×Dic612yes—
285C22×C32C2×Dic612yes—
286C22×C32C2×C3×Q812yes—
287C22×C32C23×D311yescharacteristic
288C22×C32C23×D311yescharacteristic
289C22×C32C24×C311yescharacteristic
290C32×C4—22no—
291C32×C4—22no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—22no—
295C32×C4—22no—
296C32×C4—34no—
297C32×C4—42no—
298C32×C4—42no—
299C32×C4—62no—
300C32×C4—64no—
301C32×C4C2×Dic612yes—
302C32×C4C2×Dic612yes—
303C32×C4C2×Dic612yes—
304C32×C4C2×Dic612yes—
305C32×C4C2×C3×Q812yes—
306C32×C4C2×C3×Q812yes—
307C3⋊Dic3—24no—
308C3×Dic3—24no—
309C3×Dic3—22no—
310C3×Dic3—24no—
311C3×Dic3—24no—
312C3×Dic3—22no—
313C3×Dic3—24no—
314C3×Dic3—44no—
315C3×Dic3—64no—
316C3×Dic3—64no—
317C3×Dic3—124no—
318C3×Dic3#23514yes—
319C3×Dic3C3×#4114yes—
320C4⋊Q8—92no—
321#125—92no—
322C22⋊Q8—94no—
323C4×Q8—94no—
324#117—31no—
325#117—91no—
326C2×C42—31no—
327C33#53311yescharacteristic, complemented, Sylow
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—41no—
332C23×C3C2×D3211yescharacteristic
333C23×C3C22×C3×D311yescharacteristic
334C23×C3C22×C3×D311yescharacteristic
335C3×Q8—188no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—32no—
348C2×C3×C4—34no—
349C2×C3×C4—32no—
350C2×C3×C4—34no—
351C2×C3×C4—44no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—62no—
355C2×C3×C4—62no—
356C2×C3×C4—64no—
357C2×C3×C4—64no—
358C2×C3×C4—64no—
359C2×C3×C4—68no—
360C2×C3×C4—94no—
361C2×C3×C4#62714yescomplemented
362C2×C3×C4C3×Dic614yescomplemented
363C2×C3×C4C3×Dic614yescomplemented
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C22×C3×D311yescharacteristic
367C2×C3×C4C22×C3×D311yescharacteristic
368C2×C3×C4C22×C3×D311yescharacteristic
369C2×C3×C4C22×C3×D311yescharacteristic
370C2×Dic3—24no—
371C2×Dic3—34no—
372C2×Dic3—34no—
373C2×Dic3—64no—
374C2×Dic3C22×C3×D312yes—
375C2×Dic3C22×C3×D314yes—
376C2×Dic3C22×C3×D312yes—
377Dic6—68no—
378Dic6—68no—
379C2×C32—22no—
380C2×C32—24no—
381C2×C32—22no—
382C2×C32—22no—
383C2×C32—22no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—22no—
388C2×C32—22no—
389C2×C32—41no—
390C2×C32—42no—
391C2×C32—42no—
392C2×C32—42no—
393C2×C32—44no—
394C2×C32C22×Dic611yescharacteristic
395C2×C32C22×Dic611yescharacteristic
396C2×C32C2×#23512yes—
397C2×C32D3×#4112yes—
398C2×C32C22×C3×Q811yescharacteristic
399C2×C32C2×C3×#4112yes—
400C2×Q8—94no—
401C22×C4—32no—
402C22×C4C3×D3211yescharacteristic
403C4⋊C4—32no—
404C4⋊C4—94no—
405C4⋊C4—94no—
406C4⋊C4—94no—
407C4⋊C4—92no—
408C22⋊C4—32no—
409C22⋊C4—92no—
410C42—32no—
411C42—31no—
412C42—31no—
413C42—32no—
414C42—92no—
415C22×C3—24no—
416C22×C3—22no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—44no—
428C22×C3—41no—
429C22×C3—42no—
430C22×C3C2×#62712yes—
431C22×C3C2×C3×Dic612yes—
432C22×C3C2×C3×Dic612yes—
433C22×C3C22×D3211yescharacteristic
434C22×C3C23×C3×D311yescharacteristic
435C22×C3C23×C3×D311yescharacteristic
436C3×C4—22no—
437C3×C4—22no—
438C3×C4—22no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—34no—
443C3×C4—34no—
444C3×C4—42no—
445C3×C4—42no—
446C3×C4—64no—
447C3×C4—64no—
448C3×C4—62no—
449C3×C4—64no—
450C3×C4—64no—
451C3×C4—62no—
452C3×C4—124no—
453C3×C4—184no—
454C3×C4C2×#62712yes—
455C3×C4C2×#62712yes—
456C3×C4C2×C3×Dic612yes—
457C3×C4C2×C3×Dic612yes—
458C3×C4C2×C3×Dic612yes—
459C3×C4C2×C3×Dic612yes—
460Dic3—22no—
461Dic3—24no—
462Dic3—64no—
463Dic3—64no—
464Dic3—124no—
465Dic3C3×#23514yes—
466C32—21no—
467C32—21no—
468C32—21no—
469C32—41no—
470C32#547011yescharacteristic, complemented
471C32#551711yescharacteristic, complemented
472C32C3×#53311yescharacteristic, complemented
473C23C2×C3×D3211yescharacteristic
474Q8—188no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—34no—
478C2×C4—64no—
479C2×C4—94no—
480C2×C4C3×#62714yescomplemented
481C2×C4C2×C3×D3211yescharacteristic
482C2×C4C2×C3×D3211yescharacteristic
483C2×C3—24no—
484C2×C3—22no—
485C2×C3—22no—
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—22no—
493C2×C3—42no—
494C2×C3—42no—
495C2×C3—44no—
496C2×C3—41no—
497C2×C3—42no—
498C2×C3D3×#23512yes—
499C2×C3C22×#62711yescharacteristic
500C2×C3C22×C3×Dic611yescharacteristic
501C2×C3C22×C3×Dic611yescharacteristic
502C2×C3C2×C3×#23512yes—
503C2×C3C3×D3×#4112yes—
504C22—22no—
505C22C2×C3×#62712yes—
506C22C22×C3×D3211yescharacteristic
507C4—34no—
508C4—62no—
509C4—64no—
510C4—184no—
511C4C2×C3×#62712yes—
512C4C2×C3×#62712yes—
513C3—21no—
514C3—21no—
515C3—21no—
516C3—41no—
517C3#1059504911yescharacteristic, direct product factor
518C3C3×#547011yescharacteristic, complemented
519C3C3×#551711yescharacteristic, complemented
520C2—22no—
521C2C3×D3×#23512yes—
522C2C22×C3×#62711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D32
commutator, cosocleC22×C32C24×C3
FrattiniC22C22×C3×D32
FittingC22×C33×C4C22
socleC22×C33C24
prime radicalC23×C33C23
hypercenterC22×C3×C4D32
hypocenterC32C3×#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C3×#63918C2semidirect
6. C3×C4×#627C2semidirect
7. C3×#63869C2semidirect
10. #10595049C3direct (N × H)
47. #63918C2×C3semidirect
48. C4×#627C2×C3semidirect
49. #63869C2×C3semidirect
51. C2×C33×C4Q8semidirect
178. C2×C32×C4Dic6semidirect
179. C2×C32×C4Dic6semidirect
180. C2×C32×C4C3×Q8semidirect
327. C33#533semidirect
361. C2×C3×C4#627semidirect
362. C2×C3×C4C3×Dic6semidirect
363. C2×C3×C4C3×Dic6semidirect
470. C32#5470semidirect
471. C32#5517semidirect
472. C32C3×#533semidirect
480. C2×C4C3×#627semidirect
517. C3#10595049direct (N × H)
518. C3C3×#5470semidirect
519. C3C3×#5517semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#641352
C32×#9262
C2×C32×C4×Dic32
C3×#639642
C3×#639182
C3×C4×#6272
C3×#638692
C3×#55173
C3×#54703
#105950493